The Generalized Gamma distribution as a useful RND under Heston's stochastic volatility model

TL;DR

This paper demonstrates that the Generalized Gamma distribution effectively models negatively skewed option prices under Heston's stochastic volatility model, outperforming Black-Scholes in skewed market conditions, with empirical validation on ETF data.

Contribution

It introduces the use of the Generalized Gamma distribution as a random number generator for option pricing under Heston's model, especially for negatively skewed distributions, validated with real market data.

Findings

GG distribution matches negatively skewed ETF options well

Black-Scholes model performs poorly with skewed data

Heston's model and GG distribution align closely for skewed options

Abstract

Following Boukai (2021) we present the Generalized Gamma (GG) distribution as a possible RND for modeling European options prices under Heston's (1993) stochastic volatility (SV) model. This distribution is seen as especially useful in situations in which the spot's price follows a negatively skewed distribution and hence, Black-Scholes based (i.e. the log-normal distribution) modeling is largely inapt. We apply the GG distribution as RND to modeling current market option data on three large market-index ETFs, namely the SPY, IWM and QQQ as well as on the TLT (an ETF that tracks an index of long term US Treasury bonds). The current option chain of each of the three market-index ETFs shows of a pronounced skew of their volatility `smile' which indicates a likely distortion in the Black-Scholes modeling of such option data. Reflective of entirely different market expectations, this distortion appears not to exist in the TLT option data. We provide a thorough modeling of the available option data we have on each ETF (with the October 15, 2021 expiration) based on the GG distribution and compared it to the option pricing and RND modeling obtained directly from a well-calibrated Heston's (1993) SV model (both theoretically and empirically, using Monte-Carlo simulations of the spot's price). All three market-index ETFs exhibit negatively skewed distributions which are well-matched with those derived under the GG distribution as RND. The inadequacy of the Black-Scholes modeling in such instances which involve negatively skewed distribution is further illustrated by its impact on the hedging factor, delta, and the immediate implications to the retail trader. In contrast, for the TLT ETF, which exhibits no such distortion to the volatility `smile', the three pricing models (i.e. Heston's, Black-Scholes and Generalized Gamma) appear to yield similar results.